Modelling with Functions
Overview
This section covers the modelling of practical situations using various function types, including polynomial, power, circular (trigonometric), exponential, and logarithmic functions. It also includes simple transformations and combinations of these functions, with a focus on piecewise (hybrid) functions.
1. Polynomial Functions
Definition
A polynomial function is a function that can be expressed in the form:
\[f(x) = a_n x^n + a_{n-1} x^{n-1} + ... + a_1 x + a_0\]
where \(n\) is a non-negative integer (the degree of the polynomial) and \(a_n, a_{n-1}, ..., a_0\) are constants (coefficients), with \(a_n \neq 0\).
Key Features
- Degree: The highest power of \(x\) in the polynomial.
- Leading Coefficient: The coefficient \(a_n\) of the term with the highest power.
- Zeros (Roots): The values of \(x\) for which \(f(x) = 0\).
- Turning Points: Points where the graph changes direction (local maxima or minima).
- End Behaviour: The behavior of the function as \(x\) approaches positive or negative infinity, determined by the leading term \(a_n x^n\).
Modelling with Polynomials
Polynomials can model various real-world scenarios, especially when approximating curves or relationships.
- Example: Modelling the trajectory of a projectile, approximating growth curves, or representing cost functions.
- Vertical Translation: \(f(x) + k\) (shifts the graph up by \(k\) units if \(k > 0\), down if \(k < 0\)).
- Horizontal Translation: \(f(x - h)\) (shifts the graph right by \(h\) units if \(h > 0\), left if \(h < 0\)).
- Vertical Stretch/Compression: \(af(x)\) (stretches vertically by a factor of \(|a|\) if \(|a| > 1\), compresses if \(0 < |a| < 1\); reflects across the x-axis if \(a < 0\)).
- Horizontal Stretch/Compression: \(f(bx)\) (compresses horizontally by a factor of \(|b|\) if \(|b| > 1\), stretches if \(0 < |b| < 1\); reflects across the y-axis if \(b < 0\)).
2. Power Functions
Definition
A power function is a function of the form:
\[f(x) = kx^p\]
where \(k\) is a constant and \(p\) is a real number.
Examples
- \(f(x) = x^2\) (parabola)
- \(f(x) = x^3\) (cubic)
- \(f(x) = \sqrt{x} = x^{1/2}\) (square root)
- \(f(x) = \frac{1}{x} = x^{-1}\) (reciprocal)
Modelling with Power Functions
Power functions are useful for modelling relationships where one variable is proportional to a power of another.
- Example: Modelling gravitational force (inverse square law), allometric scaling in biology.
Power functions are transformed in the same way as polynomial functions.
3. Circular (Trigonometric) Functions
Definitions
- Sine Function: \(f(x) = A\sin(B(x - C)) + D\)
- Cosine Function: \(f(x) = A\cos(B(x - C)) + D\)
- Tangent Function: \(f(x) = A\tan(B(x - C)) + D\)
Key Parameters
- Amplitude: \(|A|\) (half the distance between the maximum and minimum values).
- Period: \(\frac{2\pi}{|B|}\) (for sine and cosine), \(\frac{\pi}{|B|}\) (for tangent).
- Phase Shift: \(C\) (horizontal shift).
- Vertical Shift: \(D\) (vertical shift).
Modelling with Circular Functions
Circular functions are ideal for modelling periodic phenomena.
- Example: Modelling oscillations, waves, seasonal variations, and cyclical processes.
The parameters A, B, C, and D directly control transformations of the basic sine, cosine and tangent functions.
4. Exponential Functions
Definition
An exponential function is a function of the form:
\[f(x) = a^x\]
where \(a > 0\) and \(a \neq 1\).
\[f(x) = A a^{B(x - C)} + D\]
Key Features
- Asymptote: A horizontal asymptote at \(y = D\).
- Growth/Decay: Exponential growth if \(a > 1\), exponential decay if \(0 < a < 1\).
Modelling with Exponential Functions
Exponential functions are used to model situations involving rapid growth or decay.
- Example: Modelling population growth, radioactive decay, compound interest.
- D represents vertical translation.
- C represents horizontal translation.
- A represents vertical stretch/compression and reflection.
- B represents horizontal stretch/compression and reflection.
5. Logarithmic Functions
Definition
A logarithmic function is the inverse of an exponential function.
\[f(x) = \log_a(x)\]
where \(a > 0\) and \(a \neq 1\).
\[f(x) = A \log_a(B(x - C)) + D\]
Key Features
- Asymptote: A vertical asymptote at \(x = C\).
- Domain: \(x > C\)
- Range: All real numbers.
Modelling with Logarithmic Functions
Logarithmic functions are used to model situations where the rate of change decreases over time.
- Example: Modelling the Richter scale (earthquake magnitude), sound intensity (decibels).
- D represents vertical translation.
- C represents horizontal translation.
- A represents vertical stretch/compression and reflection.
- B represents horizontal stretch/compression and reflection.
6. Combinations of Functions
Arithmetic Combinations
Functions can be combined using arithmetic operations:
* Sum: \((f + g)(x) = f(x) + g(x)\)
* Difference: \((f - g)(x) = f(x) - g(x)\)
* Product: \((f \cdot g)(x) = f(x) \cdot g(x)\)
* Quotient: \((\frac{f}{g})(x) = \frac{f(x)}{g(x)}\), where \(g(x) \neq 0\)
Composition of Functions
\((f \circ g)(x) = f(g(x))\)
7. Piecewise (Hybrid) Functions
Definition
A piecewise function is a function defined by multiple sub-functions, each applying to a specific interval of the domain.
\[f(x) = \begin{cases} f_1(x), & \text{if } x < a \\ f_2(x), & \text{if } a \leq x < b \\ f_3(x), & \text{if } x \geq b \end{cases}\]
Modelling with Piecewise Functions
Piecewise functions are useful for modelling situations where the relationship between variables changes abruptly at certain points.
- Example: Modelling tax brackets, electricity pricing (different rates for different usage levels), step functions.
Example: Sketching a Piecewise Function
Consider the function:
\[f(x) = \begin{cases} x^2, & \text{if } x < 1 \\ 2x - 1, & \text{if } x \geq 1 \end{cases}\]
To sketch this, graph \(y = x^2\) for \(x < 1\) and \(y = 2x - 1\) for \(x \geq 1\).
Summary
Understanding and applying transformations to polynomial, power, circular, exponential, and logarithmic functions, along with creating and interpreting piecewise functions, is crucial for modelling various practical situations in Mathematical Methods. This includes sketching graphs, determining function rules, and solving related problems.
